CN111929892A - Off-axis multi-reflector system design method based on space coordinate transfer matrix - Google Patents

Off-axis multi-reflector system design method based on space coordinate transfer matrix Download PDF

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CN111929892A
CN111929892A CN202010862358.0A CN202010862358A CN111929892A CN 111929892 A CN111929892 A CN 111929892A CN 202010862358 A CN202010862358 A CN 202010862358A CN 111929892 A CN111929892 A CN 111929892A
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mirror
coordinate system
reflector
spatial
transfer matrix
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CN111929892B (en
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程同蕾
陈晓雨
赵继
李曙光
王方
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Northeastern University China
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Abstract

The invention discloses a method for designing an off-axis multi-reflector optical system based on a space coordinate transfer matrix, and belongs to the technical field of optical design. The method establishes an initial coordinate system and a reference coordinate system of each reflector; selecting characteristic light rays, establishing an input-output relation among the reflectors, and recording intersection points of the characteristic light rays and the reflectors as characteristic points; the transmission of the light ray characteristics in the off-axis multi-reflector optical system is realized by a method of a space coordinate transmission matrix; and finally, fitting to obtain the pose of each reflector of the off-axis multi-reflector optical system. The invention designs the optical system by using the space coordinate transfer matrix theory, and is beneficial to the integrated design and processing of the optical system and the assembly and adjustment of the system.

Description

Off-axis multi-reflector system design method based on space coordinate transfer matrix
Technical Field
The invention relates to the technical field of optical design, in particular to a method for designing an off-axis multi-reflector optical system based on a space coordinate transfer matrix.
Background
With the increasing progress of space optical remote sensing and aerospace technologies such as rockets, satellites and space stations, various design and processing technical indexes of optical systems are higher and higher, and the requirements of high resolution and wide coverage range are increasingly urgent. However, the conflicting relationship between high resolution and large field of view limits its development to some extent. Early optical systems all adopt refractive systems, and with the research of people on optical design theory and reflective systems, catadioptric systems and all-trans systems gradually appear. Higher resolution and less energy loss then become an issue to be addressed, thus creating an off-axis optical system.
One conventional design method for off-axis optical systems is to take existing patents or other available systems as an initial structure and further optimize them with optical design software to obtain the final design result. Another common design method is to create an initial structure of a coaxial spherical surface or a conical surface according to a three-level (or seidel) aberration theory, then obtain a non-blocking structure by a diaphragm off-axis, a field off-axis or an off-axis manner of tilting a mirror, and finally optimize by using optical design software to obtain a final structure. However, the difficulty of processing and adjusting the system of the off-axis optical element is high, and the processing precision of the element and the adjusting method of the system are key factors for restricting the further development of the element.
The space coordinate transfer matrix is proposed based on a matrix method to describe the kinematics and dynamics of the robot manipulator. The mathematical description transforms a four-order square matrix into homogeneous coordinates of three-dimensional spatial points, and the motion, transformation and mapping are associated with matrix operations and performed as a unified body in the processes of machining or assembly and the like. The invention is provided under the background that the optical system design is carried out by utilizing the space coordinate transfer matrix theory, which is beneficial to the integrated design and processing of the optical system and the assembly and adjustment of the system.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a method for designing an off-axis multi-reflecting optical system based on a space coordinate transfer matrix, which is used for realizing the integrated design, processing and system adjustment of the optical system.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a method for designing an off-axis multi-reflector optical system based on a space coordinate transfer matrix comprises the following steps:
step 1: creating M mirrors M in an off-axis multi-mirror optical system1、M2......MmThe local coordinate system and the reference coordinate system;
m reflectors M in the off-axis multi-reflector optical system1、M2......MmThe local coordinate system takes the center of each reflector as the origin, and the reference coordinate system takes the first reflector M1The center of (2) is an origin, and an X axis, a Y axis and a Z axis are determined according to a right-hand rule.
Step 2: selecting n characteristic light rays L1、L2、L3......LnN characteristic light rays intersect with M reflectors and are respectively provided with n characteristic points, M1The characteristic points in (1) are recorded as: s11、S12、S13......S1n，M2The characteristic point in (1) is marked as S21、S22、S23......S2n，MmThe characteristic points in (1) are recorded as: sm1、Sm2、Sm3......Smn；
And step 3: according to the law of reflection, the first mirror M1As a second mirror M2Input of, a second mirror M2As the output of the third mirror M3By analogy, M-1 th mirror Mm-1As the M-th mirror MmThe input of (1);
and 4, step 4: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000021
Second reflector M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA0002648561130000022
By analogy, the M-th reflector M is calculatedmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure BDA0002648561130000023
Process for exampleThe following:
step 4.1: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000024
The formula of (1) is as follows:
Figure BDA0002648561130000025
wherein, theta1For referencing the reference coordinate system1The rotation angle of the spatial local coordinate system,
Figure BDA0002648561130000026
is that
Figure BDA0002648561130000027
Figure BDA0002648561130000028
The three are randomly arranged and combined;
due to the first reflector M1Is the same as the origin of the reference coordinate system, then
Figure BDA0002648561130000029
Step 4.2: calculating the second mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA00026485611300000210
The formula of (1) is as follows:
Figure BDA00026485611300000211
wherein, theta2To use the first reflector M1The spatial local coordinate system of the second reflector M is a reference2The rotation angle of the spatial local coordinate system,
Figure BDA00026485611300000212
is that
Figure BDA00026485611300000213
Figure BDA00026485611300000214
The three are randomly arranged and combined;
due to the second reflector M2The spatial local coordinate system and the first reflector M1The origin of the spatial local coordinate system of (a) is not the same:
Figure BDA0002648561130000031
is that
Figure BDA0002648561130000032
One of 7 kinds; the values of x, y and z may be any of various values.
Step 4.3: calculating the third mirror M3To the second mirror M2Spatial coordinate transfer matrix of
Figure BDA0002648561130000033
With reference to step 4.2, the same applies until the M-th mirror M is calculatedmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure BDA0002648561130000034
The M-th reflecting mirror MmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure BDA0002648561130000035
The formula of (1) is as follows:
Figure BDA0002648561130000036
wherein, thetamIs provided with an M-1 th reflecting mirror Mm-1The spatial local coordinate system of (a) is a referenceM mirror MmThe rotation angle of the spatial local coordinate system,
Figure BDA0002648561130000037
is that
Figure BDA0002648561130000038
Figure BDA0002648561130000039
The three are randomly arranged and combined;
due to M-th reflecting mirror MmAnd the M-1 th mirror Mm-1The origin of the spatial coordinate system of (a) is not the same:
Figure BDA00026485611300000310
is that
Figure BDA00026485611300000311
And one of the above-mentioned 7, the above-mentioned x, y and z can take any different values.
In said step 4.1
Figure BDA00026485611300000312
The random permutation and combination mode of the three is as follows:
first reflector M1Relative to a reference coordinate system X0、Y0、Z0One of the shafts rotates axially, i.e. has
Figure BDA00026485611300000313
(ii) a condition;
first reflector M1Relative to a reference coordinate system X0、Y0、Z0The two shafts rotating successively, i.e. having
Figure BDA00026485611300000314
(ii) a condition;
first reflector M1Relative to the local coordinate system ofIn a reference coordinate system X0、Y0、Z0Three of the shafts rotate in sequence, i.e.
Figure BDA00026485611300000315
(ii) a condition;
thus, the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000041
Is provided with
Figure BDA0002648561130000042
(ii) a condition;
in said step 4.2
Figure BDA0002648561130000043
The random permutation and combination mode of the three is as follows:
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1One of the shafts rotates axially, i.e. has
Figure BDA0002648561130000044
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1The two shafts rotating successively, i.e. having
Figure BDA0002648561130000045
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1Three of the shafts rotate in sequence, i.e.
Figure BDA0002648561130000046
(ii) a condition;
thus, the second inverseBeam mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA0002648561130000047
Is provided with
Figure BDA0002648561130000048
(ii) a condition;
in said step 4.3
Figure BDA0002648561130000049
The random permutation and combination mode of the three is as follows:
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1One of the shafts rotates axially, i.e. has
Figure BDA00026485611300000410
(ii) a condition;
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1The two shafts rotating successively, i.e. having
Figure BDA00026485611300000411
(ii) a condition;
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1Three of the shafts rotate in sequence, i.e.
Figure BDA00026485611300000412
(ii) a condition;
therefore, the M-th mirror MmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure BDA00026485611300000413
Is provided with
Figure BDA00026485611300000414
And (3) a situation.
The spatial coordinate transfer matrix
Figure BDA00026485611300000415
The product of (a) is the spatial coordinate transfer matrix T of the optical system:
Figure BDA00026485611300000416
the system space coordinate transfer matrix T of the optical system has 15 x 105n-1And (3) a situation.
And 5: according to the space coordinate transfer matrix, the description of the characteristic point coordinates in the m reflectors in a reference coordinate system is calculated, and the process is as follows:
step 5.1: transferring matrices according to spatial coordinates
Figure BDA00026485611300000417
Calculating the first mirror M1Characteristic point S of11、S12、S13......S1nDescription in a reference coordinate System
Figure BDA0002648561130000051
Figure BDA0002648561130000052
Wherein, P1Is a first reflector M1Each characteristic point S of11、S12、S13......S1nAt the first reflector M1Description of coordinates in a local coordinate system;
step 5.2: transferring matrices according to spatial coordinates
Figure BDA0002648561130000053
Calculating the second mirror M2Characteristic point S of21、S22、S23......S2nRoot of Henren ginsengDescription in a reference frame
Figure BDA0002648561130000054
Figure BDA0002648561130000055
Wherein, P2Is the second mirror M2Each characteristic point S of21、S22、S23......S2nAt the second mirror M2Description of coordinates in a local coordinate system;
step 5.3: and so on until the matrix is transferred according to the spatial coordinates
Figure BDA0002648561130000056
Calculate mth mirror MmCharacteristic point S ofm1、Sm2、Sm3......SmnDescription in a reference coordinate System
Figure BDA0002648561130000057
Figure BDA0002648561130000058
Wherein, PmIs the M-th reflecting mirror MmEach characteristic point S ofm1、Sm2、Sm3......SmnAt M-th reflecting mirror MmDescription of coordinates in a local coordinate system.
The first reflector M in step 5.11Each characteristic point S of11、S12、S13......S1nDescription of coordinates in a reference coordinate system
Figure BDA0002648561130000059
All the possibilities of
Figure BDA00026485611300000510
In one case, 15 × n cases in total;
second mirror M in said step 5.22Each characteristic point S of21、S22、S23......S2nDescription of coordinates in a reference coordinate system
Figure BDA00026485611300000511
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
by analogy, the third reflector M3Each characteristic point S of31、S32、S33......S3nDescription of coordinates in a reference coordinate system
Figure BDA00026485611300000512
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
m-th reflecting mirror M in the step 5.3mEach characteristic point S ofm1、Sm2、Sm3......SmnDescription of coordinates in a reference coordinate system
Figure BDA00026485611300000513
All of the possibilities of (1) are 15 x 105n-1In one case, 15 × 105 in totaln-1N cases.
Step 6: and fitting to obtain the poses of the m reflectors of the off-axis multi-reflector optical system according to the coordinate description of the characteristic points in the m reflectors in the reference coordinate system.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in:
1. the method provided by the invention establishes M reflectors M in establishing an off-axis multi-reflector optical system1、M2......MmThe local coordinate system and the reference coordinate system embody the characteristic light propagation path by a coordinate language, are convenient for modeling in different optical design and machining software, and achieve the best design and machining and error analysis effects;
2. the local coordinate system and the reference coordinate system of each reflector use a common right-hand criterion, so that the transmission calculation of the coordinates of the characteristic points in the space coordinate system is facilitated, and the solution of the characteristic light rays in different coordinate planes is facilitated.
3. By selecting representative characteristic light rays for marking and calculating, the processing process of a large amount of light ray data is simplified, digitalized and highlighted, the working efficiency is improved, and the design amount of scientific research workers is reduced.
4. And (3) establishing input and output relations among the reflectors by using a reflection law to form an integrated analysis framework, wherein the integrated analysis framework is clear and understandable compared with other optical theories and can be accepted by optical design and machining manufacturers at various stages.
5. The concept of the space coordinate transfer matrix is introduced into the optical design, the leading idea of the mechanical processing is introduced into the optical design field, the creation of an initial model of the optical design is guided and constrained, the unification of the two stages of theories is beneficial to the system design and the processing and debugging, and the better fusion development of the two fields is promoted.
6. The analysis of different conditions of the space coordinate transfer matrix provides a large amount of freedom degrees for the design of a space optical system, different optical systems can be obtained through the combination of different conditions, the steps of designing the optical systems are greatly simplified, and the working efficiency is improved.
7. Due to the establishment of the local coordinate system of the individual mirrors, the local coordinates of the characteristic points marked on the basis of the characteristic rays can be determined. The spatial coordinate transfer matrix establishes the relation between the reflector and other reflectors, and the relation between each component in the optical system is highlighted by adopting a method of multiplying the spatial coordinate transfer matrix, so that the description of the characteristic point under a reference coordinate system can be obtained, and the description under different local coordinate systems can also be obtained.
8. After the coordinate description of the characteristic points in each reflector under the reference coordinate system is known, namely when all the characteristic points are in one coordinate measurement, the poses of m reflectors of different types of reflectors can be obtained through multiple fitting modes, and then different optical systems can be obtained.
In summary, the off-axis multi-reflector optical system design method based on the spatial coordinate transfer matrix provided by the invention associates the machining with the optical design, and is beneficial to mutual fusion, mutual restriction and mutual guidance of the two fields. Different optical systems can be obtained by adopting different combination modes of space coordinate transfer matrixes, and then different types of optical systems can also be obtained by adopting different characteristic point coordinate fitting modes. Therefore, a great deal of freedom is given to the design process, and a novel optical system design idea is provided.
Drawings
FIG. 1 is a flow chart of a method for designing an off-axis multi-mirror optical system based on a spatial coordinate transfer matrix according to an embodiment of the present invention;
FIG. 2 is a diagram of an example of an off-axis three-mirror optical system according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of the transformation of the spatial coordinate system of the off-axis three-mirror optical system according to the embodiment of the present invention.
Detailed Description
The following detailed description of embodiments of the present invention is provided in connection with the accompanying drawings and examples. The following examples are intended to illustrate the invention but are not intended to limit the scope of the invention.
The flow of the off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix in this embodiment is shown in fig. 1, and includes the following steps:
step 1: in this embodiment, as shown in fig. 2 and 3, 3 reflectors M in the off-axis multi-mirror optical system are established1、M2、M3The local coordinate system and the reference coordinate system;
3 reflectors M in the off-axis multi-reflector optical system1、M2、M3The local coordinate system takes the center of each reflector as an origin O1、O2And O3Reference coordinate system with a first mirror M1Center O of1Is the origin O, i.e. the origins O and O of the reference coordinate system1And (5) overlapping, and determining an X axis, a Y axis and a Z axis according to a right-hand rule.
Step 2: selecting n characteristic light rays L1、L2、L3......LnN characteristic light rays intersect with M reflectors and are respectively provided with n characteristic points, M1The characteristic points in (1) are recorded as: s11、S12、S13......S1n，M2The characteristic point in (1) is marked as S21、S22、S23......S2n，M3The characteristic points in (1) are recorded as: s31、S32、S33......S3n；
And step 3: according to the law of reflection, the first mirror M1As a second mirror M2Input of, a second mirror M2As the output of the third mirror M3The input of (1);
and 4, step 4: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000071
Second reflector M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA0002648561130000072
Calculating the 3 rd mirror M3To the 2 nd mirror M2Spatial coordinate transfer matrix of
Figure BDA0002648561130000073
The process is as follows:
step 4.1: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000074
The formula of (1) is as follows:
Figure BDA0002648561130000075
wherein, theta1For referencing the reference coordinate system1The rotation angle of the spatial local coordinate system,
Figure BDA0002648561130000076
is that
Figure BDA0002648561130000077
Figure BDA0002648561130000078
The three are randomly arranged and combined;
due to the first reflector M1Is the same as the origin of the reference coordinate system, then
Figure BDA0002648561130000081
Step 4.2: calculating the second mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA0002648561130000082
The formula of (1) is as follows:
Figure BDA0002648561130000083
wherein, theta2To use the first reflector M1The spatial local coordinate system of the second reflector M is a reference2The rotation angle of the spatial local coordinate system,
Figure BDA0002648561130000084
is that
Figure BDA0002648561130000085
Figure BDA0002648561130000086
The three are randomly arranged and combined;
due to the second reflector M2The spatial local coordinate system and the first reflector M1The origin of the spatial local coordinate system of (a) is not the same:
Figure BDA0002648561130000087
is that
Figure BDA0002648561130000088
One of 7 kinds; x is the number of2、y2、z2Is the second mirror M2Relative to the first mirror M1Of the spatial local coordinate system, x2、y2、z2Any of various values may be used.
Step 4.3: calculating the third mirror M3To the second mirror M2Spatial coordinate transfer matrix of
Figure BDA00026485611300000813
The formula of (1) is as follows:
Figure BDA0002648561130000089
wherein, theta3To use a 2 nd reflecting mirror M2The spatial local coordinate system of (3) is a reference reflection mirror M3The rotation angle of the spatial local coordinate system,
Figure BDA00026485611300000810
is that
Figure BDA00026485611300000811
Figure BDA00026485611300000812
The three are randomly arranged and combined;
due to the third reflector M3Space coordinate system and second reflector M2The origin of the spatial coordinate system of (a) is not the same:
Figure BDA0002648561130000091
is that
Figure BDA0002648561130000092
One of 7 kinds. x is the number of3、y3、z3Is a third reflector M3Relative to the second mirror M2Of the spatial local coordinate system, x3、y3、z3Any of various values may be used.
In this embodiment, the first reflector M is based on the reference coordinate system1Rotation angle theta of a spatial local coordinate system1With the first reflector M1The spatial local coordinate system of the second reflector M is a reference2Rotation angle theta of a spatial local coordinate system2With a second mirror M2The spatial local coordinate system of (a) is a reference third reflector M3Rotation angle theta of a spatial local coordinate system3Is the same rotation angle and is marked as theta, wherein the coordinate system is O'2-X'2Y'2Z'2、O'3-X'3Y'3Z'3Represents the second mirror M2Relative to the first mirror M1A transitional local coordinate system in which the local coordinate system is rotated by theta and then translated, and a third reflector M3Local coordinate system relative to the second mirror M2The local coordinate system of (2) is rotated by theta and then translated by the transitional local coordinate system, as shown in fig. 3.
In said step 4.1
Figure BDA0002648561130000093
The random permutation and combination mode of the three is as follows:
first reflector M1Relative to a reference coordinate system X0、Y0、Z0One of the shafts rotates axially, i.e. has
Figure BDA0002648561130000094
(ii) a condition;
first reflector M1Relative to a reference coordinate system X0、Y0、Z0The two shafts rotating successively, i.e. having
Figure BDA0002648561130000095
(ii) a condition;
first reflector M1Relative to a reference coordinate system X0、Y0、Z0Three axial parts in the shaftDo not rotate in sequence, i.e. have
Figure BDA0002648561130000096
(ii) a condition;
thus, the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure BDA0002648561130000097
Is provided with
Figure BDA0002648561130000098
(ii) a condition;
in said step 4.2
Figure BDA0002648561130000099
The random permutation and combination mode of the three is as follows:
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1One of the shafts rotates axially, i.e. has
Figure BDA00026485611300000910
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1The two shafts rotating successively, i.e. having
Figure BDA00026485611300000911
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1Three of the shafts rotate in sequence, i.e.
Figure BDA00026485611300000912
(ii) a condition;
thus, the second mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure BDA0002648561130000101
Is provided with
Figure BDA0002648561130000102
(ii) a condition;
in said step 4.3
Figure BDA0002648561130000103
The random permutation and combination mode of the three is as follows:
third reflector M3Relative to the second mirror M2Local coordinate system X2、Y2、Z2One of the shafts rotates axially, i.e. has
Figure BDA0002648561130000104
(ii) a condition;
third reflector M3Relative to the second mirror M2Local coordinate system X2、Y2、Z2The two shafts rotating successively, i.e. having
Figure BDA0002648561130000105
(ii) a condition;
third reflector M3Relative to the second mirror M2Local coordinate system X2、Y2、Z2Three of the shafts rotate in sequence, i.e.
Figure BDA0002648561130000106
(ii) a condition;
thus, the third mirror M3To the second mirror M2Spatial coordinate transfer matrix of
Figure BDA0002648561130000107
Is provided with
Figure BDA0002648561130000108
And (3) a situation.
The spatial coordinate transfer matrix
Figure BDA0002648561130000109
The product of (a) is the spatial coordinate transfer matrix T of the optical system:
Figure BDA00026485611300001010
there are 15 × 105 cases of the system space coordinate transfer matrix T of the optical system.
And 5: according to the space coordinate transfer matrix, the description of the characteristic point coordinates in the 3 reflectors in the reference coordinate system is calculated, and the process is as follows:
step 5.1: transferring matrices according to spatial coordinates
Figure BDA00026485611300001011
Calculating the first mirror M1Characteristic point S of11、S12、S13......S1nDescription in a reference coordinate System
Figure BDA00026485611300001012
Figure BDA00026485611300001013
Wherein, P1Is a first reflector M1Each characteristic point S of11、S12、S13......S1nAt the first reflector M1Description of coordinates in a local coordinate system;
step 5.2: transferring matrices according to spatial coordinates
Figure BDA00026485611300001014
Calculating the second mirror M2Characteristic point S of21、S22、S23......S2nDescription in a reference coordinate System
Figure BDA00026485611300001015
Figure BDA00026485611300001016
Wherein, P2Is the second mirror M2Each characteristic point S of21、S22、S23......S2nAt the second mirror M2Description of coordinates in a local coordinate system;
step 5.3: and so on until the matrix is transferred according to the spatial coordinates
Figure BDA0002648561130000111
Calculating the 3 rd mirror M3Characteristic point S of31、S32、S33......S3nDescription in a reference coordinate System
Figure BDA0002648561130000112
Figure BDA0002648561130000113
Wherein, P3Is the 3 rd reflecting mirror M3Each characteristic point S of31、S32、S33......S3nAt the 3 rd mirror M3Description of coordinates in a local coordinate system.
The first reflector M in step 5.11Each characteristic point S of11、S12、S13......S1nDescription of coordinates in a reference coordinate system
Figure BDA0002648561130000114
All the possibilities of
Figure BDA0002648561130000115
In one case, 15 × n cases in total;
second mirror M in said step 5.22Each characteristic point S of21、S22、S23......S2nDescription of coordinates in a reference coordinate system
Figure BDA0002648561130000116
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
by analogy, the third reflector M3Each characteristic point S of31、S32、S33......S3nDescription of coordinates in a reference coordinate system
Figure BDA0002648561130000117
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
step 6: and fitting to obtain the poses of the 3 reflectors of the off-axis multi-reflector system according to the coordinate description of the characteristic points in the 3 reflectors in the reference coordinate system.

Claims (8)

1. A method for designing an off-axis multi-reflector optical system based on a space coordinate transfer matrix is characterized by comprising the following steps:
step 1: creating M mirrors M in an off-axis multi-mirror optical system1、M2......MmThe local coordinate system and the reference coordinate system;
step 2: selecting n characteristic light rays L1、L2、L3......LnN characteristic light rays intersect with M reflectors and are respectively provided with n characteristic points, M1The characteristic points in (1) are recorded as: s11、S12、S13......S1n，M2The characteristic point in (1) is marked as S21、S22、S23......S2n，MmThe characteristic points in (1) are recorded as: sm1、Sm2、Sm3......Smn；
And step 3: according to the law of reflection, the first mirror M1As a second mirror M2Input of, a second mirror M2As the output of the third mirror M3By analogy, M-1 th mirror Mm-1As the mth inverseBeam mirror MmThe input of (1);
and 4, step 4: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure FDA0002648561120000011
Second reflector M2To the first reflector M1Spatial coordinate transfer matrix of
Figure FDA0002648561120000012
By analogy, the M-th reflector M is calculatedmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure FDA0002648561120000013
And 5: according to the space coordinate transfer matrix, calculating the description of the characteristic point coordinates in the m reflectors in a reference coordinate system;
step 6: and fitting to obtain the poses of the m reflectors of the off-axis multi-reflector optical system according to the coordinate description of the characteristic points in the m reflectors in the reference coordinate system.
2. The off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 1, wherein: m reflectors M in the off-axis multi-reflector optical system1、M2......MmThe local coordinate system takes the center of each reflector as the origin, and the reference coordinate system takes the first reflector M1The center of (2) is an origin, and an X axis, a Y axis and a Z axis are determined according to a right-hand rule.
3. The off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 1, wherein: the process of the step 4 is as follows:
step 4.1: calculating the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure FDA0002648561120000014
The formula of (1) is as follows:
Figure FDA0002648561120000015
wherein, theta1For referencing the reference coordinate system1The rotation angle of the spatial local coordinate system,
Figure FDA0002648561120000016
is that
Figure FDA0002648561120000021
Figure FDA0002648561120000022
The three are randomly arranged and combined;
due to the first reflector M1Is the same as the origin of the reference coordinate system, then
Figure FDA0002648561120000023
Step 4.2: calculating the second mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure FDA0002648561120000024
The formula of (1) is as follows:
Figure FDA0002648561120000025
wherein, theta2To use the first reflector M1The spatial local coordinate system of the second reflector M is a reference2The rotation angle of the spatial local coordinate system,
Figure FDA0002648561120000026
is that
Figure FDA0002648561120000027
Figure FDA0002648561120000028
The three are randomly arranged and combined;
due to the second reflector M2The spatial local coordinate system and the first reflector M1The origin of the spatial local coordinate system of (a) is not the same:
Figure FDA0002648561120000029
is that
Figure FDA00026485611200000210
One of the above, wherein x, y, z are displacement amounts of a spatial coordinate system;
step 4.3: calculating the third mirror M3To the second mirror M2Spatial coordinate transfer matrix of
Figure FDA00026485611200000211
With reference to step 4.2, the same applies until the M-th mirror M is calculatedmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure FDA00026485611200000212
The M-th reflecting mirror MmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure FDA00026485611200000213
The formula of (1) is as follows:
Figure FDA00026485611200000214
wherein, thetamIs provided with an M-1 th reflecting mirror Mm-1The spatial local coordinate system of (a) is a reference M-th reflecting mirror MmThe rotation angle of the spatial local coordinate system,
Figure FDA0002648561120000031
is that
Figure FDA0002648561120000032
Figure FDA0002648561120000033
The three are randomly arranged and combined;
due to M-th reflecting mirror MmAnd the M-1 th mirror Mm-1The origin of the spatial coordinate system of (a) is not the same:
Figure FDA0002648561120000034
is that
Figure FDA0002648561120000035
Wherein x, y, z are displacement amounts of a spatial coordinate system.
4. The off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 3, wherein:
in said step 4.1
Figure FDA0002648561120000036
The random permutation and combination mode of the three is as follows:
first reflector M1Relative to a reference coordinate system X0、Y0、Z0One of the shafts rotates axially, i.e. has
Figure FDA0002648561120000037
(ii) a condition;
first reflector M1Relative to a reference coordinate system X0、Y0、Z0The two shafts rotating successively, i.e. having
Figure FDA0002648561120000038
(ii) a condition;
first reflector M1Relative to a reference coordinate system X0、Y0、Z0Three of the shafts rotate in sequence, i.e.
Figure FDA0002648561120000039
(ii) a condition;
thus, the first mirror M1Spatial coordinate transfer matrix to a reference coordinate system
Figure FDA00026485611200000310
Is provided with
Figure FDA00026485611200000311
(ii) a condition;
in said step 4.2
Figure FDA00026485611200000312
The random permutation and combination mode of the three is as follows:
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1One of the shafts rotates axially, i.e. has
Figure FDA00026485611200000313
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1The two shafts rotating successively, i.e. having
Figure FDA00026485611200000314
(ii) a condition;
second reflector M2Relative to the first mirror M1Local coordinate system X1、Y1、Z1Three of the shafts rotate in sequence, i.e.
Figure FDA00026485611200000315
(ii) a condition;
thus, the second mirror M2To the first reflector M1Spatial coordinate transfer matrix of
Figure FDA0002648561120000041
Is provided with
Figure FDA0002648561120000042
(ii) a condition;
in said step 4.3
Figure FDA0002648561120000043
The random permutation and combination mode of the three is as follows:
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1One of the shafts rotates axially, i.e. has
Figure FDA0002648561120000044
(ii) a condition;
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1The two shafts rotating successively, i.e. having
Figure FDA0002648561120000045
(ii) a condition;
m-th reflecting mirror MmRelative to the M-1 th mirror Mm-1Local coordinate system Xm-1、Ym-1、Zm-1Three of the shafts rotate in sequence, i.e.
Figure FDA0002648561120000046
(ii) a condition;
therefore, the M-th mirror MmTo M-1 th mirror Mm-1Spatial coordinate transfer matrix of
Figure FDA0002648561120000047
Is provided with
Figure FDA0002648561120000048
And (3) a situation.
5. The off-axis multi-reflector system design method based on the spatial coordinate transfer matrix as claimed in claim 3 or 4, wherein: the spatial coordinate transfer matrix
Figure FDA0002648561120000049
The product of (a) is the spatial coordinate transfer matrix T of the optical system:
Figure FDA00026485611200000410
6. the off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 5, wherein: the system space coordinate transfer matrix T of the optical system has 15 x 105n-1And (3) a situation.
7. The off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 1, wherein: the process of the step 5 is as follows:
step 5.1: transferring matrices according to spatial coordinates
Figure FDA00026485611200000411
Calculating the first mirror M1Characteristic point S of11、S12、S13......S1nDescription in a reference coordinate System
Figure FDA00026485611200000412
Figure FDA00026485611200000413
Wherein, P1Is a first reflector M1Each characteristic point S of11、S12、S13......S1nAt the first reflector M1Description of coordinates in a local coordinate system;
step 5.2: transferring matrices according to spatial coordinates
Figure FDA00026485611200000414
Calculating the second mirror M2Characteristic point S of21、S22、S23......S2nDescription in a reference coordinate System
Figure FDA00026485611200000415
Figure FDA0002648561120000051
Wherein, P2Is the second mirror M2Each characteristic point S of21、S22、S23......S2nAt the second mirror M2Description of coordinates in a local coordinate system;
step 5.3: and so on until the matrix is transferred according to the spatial coordinates
Figure FDA0002648561120000052
Calculate mth mirror MmCharacteristic point S ofm1、Sm2、Sm3......SmnDescription in a reference coordinate System
Figure FDA0002648561120000053
Figure FDA0002648561120000054
Wherein, PmIs the M-th reflecting mirror MmEach characteristic point S ofm1、Sm2、Sm3......SmnAt M-th reflecting mirror MmDescription of coordinates in a local coordinate system.
8. The off-axis multi-mirror optical system design method based on the spatial coordinate transfer matrix as claimed in claim 7, wherein:
the first reflector M in step 5.11Each characteristic point S of11、S12、S13......S1nDescription of coordinates in a reference coordinate system
Figure FDA0002648561120000055
All the possibilities of
Figure FDA0002648561120000056
In one case, 15 × n cases in total;
second mirror M in said step 5.22Each characteristic point S of21、S22、S23......S2nDescription of coordinates in a reference coordinate system
Figure FDA0002648561120000057
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
by analogy, the third reflector M3Each characteristic point S of31、S32、S33......S3nDescription of coordinates in a reference coordinate system
Figure FDA0002648561120000058
There are 15 × 105 cases, for a total of 15 × 105 × n cases;
in said step 5.3M-th reflecting mirror MmEach characteristic point S ofm1、Sm2、Sm3......SmnDescription of coordinates in a reference coordinate system
Figure FDA0002648561120000059
All of the possibilities of (1) are 15 x 105n-1In one case, 15 × 105 in totaln-1N cases.
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